A SunCam online continuing education course
Basic Trigonometry, Significant Figures,
and Rounding
A Quick Review
(Free of Charge and Not for Credit)
by
Professor Patrick L. Glon, P.E.
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review is a review of
some fundamental principles of basic math for use by engineers. This course is prepared for
those who might find themselves a bit rusty and would like a quick refresher.
The information in the course is useful for application to the solution of structural problems
especially in the fields of statics and strength of materials.
The trigonometry review includes demonstrating - through the use of several example problems
– the use of the basic trigonometric functions including:







the sine (and its inverse, sin-1)
the cosine (and its inverse, cos-1)
the tangent (and its inverse, tan-1)
the Pythagorean Theorem
the Sum of the Angles
the Law of Sines; and
the Law of Cosines
The significant figures and rounding review includes a discussion of the precision and validity of
an answer, along with rules and guidelines for using the appropriate number of significant
figures, and for rounding answers appropriately.
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Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 2 of 10
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
Trigonometry Review
All triangles have six values – three side lengths and three interior angles. And all triangle
problems must have at least three known values and at most three unknown values. The solution
to a triangle problem – or to solve a triangle problem – is to find the unknown values.
Right Triangle – A right triangle is simply a triangle where one of the angles is 90°. Since one
angle is known, 90°, only two other values must be known to solve the triangle problem. And of
those two other known values, at least one must be the length of a side. A triangle with three
known angles – and not a length of a side – can’t be solved because there are an infinite number
of triangles that can be drawn when only the angles are known – from very small triangles to
very big triangles. They will all be shaped the same.
Right-Triangle Trigonometric Relationships – For the right triangle shown below, angle A is
the right angle, the 90° angle. The side opposite the 90° angle (identified by the corresponding
lower case letter “a”) is called the hypotenuse (hyp) of the triangle.
The other two angles of the triangle (“B” and “C”) also each have an opposite side. The opposite
side of an angle is usually identified with the same letter of the angle only in the opposite case –
in this case “b”, and “c”. The angle and the side opposite are usually the same letter – one is a
capital letter and the other is a lower case letter. In the triangle below, we used the upper case as
the angle and, therefore, the lower case as the side opposite.
In a right triangle, the two angles that are not 90° have an adjacent (adj) and an opposite (opp)
side. For angle B, the opposite side is “b”, and the adjacent side is “c”. For angle C, the
opposite side is “c” and the adjacent side is “b”.
Trigonometric Relationships
sin
cos
Pythagorean
Theorem
a2 = b2 + c2 Sum of the Angles
A + B + C = 180° B + C = 90° tan
Right Triangle
To solve a right triangle, the right angle trigonometric relationships, the Pythagorean Theorem,
and the Sum of the Angles are needed. These are very important relationships and are shown
above. They apply to the right triangle shown above.
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Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 3 of 10
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
Often the right angle is not noted as “A = 90°” as shown above. It is common practice to simply
draw a small “box” in the corner of the right angle as shown in the triangle below. It is
understood that the angle is 90°.
Example: Find the unknown values of the right triangle if the known values are as shown below. A + B + C = 180° Sin 65° = opp / hyp = b/12 C = 180° ‐ A – B b = (12’) x (Sin 65°) C = 180 – 90 – 65 b = (12) x (0.9063) C = 35° b = 10.9’ a2 = b2 + c2 2
2
2
c = a – b c2 = (12)2 – (10.9)2 c2 = 144 – 119 c2 = 25 c = 5’ Scalene Triangle – A scalene triangle is a triangle that does NOT have a 90° angle. All three
sides are usually different lengths (but don’t have to be). Of the three known values, at least one
must be the length of a side because, again, a triangle with only three known angles cannot be
solved.
Scalene Triangle Trigonometric Relationships – In addition to the trigonometric relationships
for the right triangle, frequently the law of sines and the law of cosines will be required to solve
for the three unknowns.
Law of Sines – The law of sines states that the ratio of the length of a side to the sine of the
opposite angle is a constant.
Law of Cosines – The law of cosines is stated below. Notice that if C = 90°, then Cos C = 0,
and the equation becomes the Pythagorean Theorem.
c2 = a2 + b2 – 2 a b Cos C
The equation can also be written as:
a2 = b2 + c2 – 2 b c Cos A
b2 = a2 + c2 – 2 a c Cos B
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Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 4 of 10
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
The following examples are triangle solutions for different combinations of known values.
Example: Find the unknown values of the triangle if the known values are in an angle-sideangle sequence.
C = 180° - 50° - 30° = 100°
6
sin 100
sin 50
sin 50
6
sin 100
sin 30
sin 100
6
sin 30
0.7660
6
0.9848
4.67
0.500
6
0.9848
3.05
Example: Find the unknown values of the triangle if the known values are in a side-angle-side
sequence.
2
4
6
2 4 6 60 16 36 – 24 28 √28 a = 5.29 0.866
60
0.1637 5.29
6
4
5.29
6 0.1637 0.9822 0.9822 C = 79.1° 4 0.1637 0.6548 0.6548 B = 40.9° 60° + 79.1° + 40.9° = 180° check  www.SunCam.com
Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 5 of 10
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
Example: Find the unknown values of the triangle if the known values are a side-side-side
sequence.
2
4
6 2 5 4
5
40
5 5
40
C 82.8° 82.8
6
5
0.992
6
0.1654
0.1654
A = 55.8°
5 0.1654
B = 41.4°
4 0.1654
82.8° + 55.8° + 41.4° = 180° check 
Example: Find the unknown values of the triangle if the known values are a side-side-angle
sequence. With these three known values there can be three different outcomes when solving for
the unknown values
Outcome 1 – there can be no solution. In the example below, the leg opposite the 30° angle is
too short to reach the leg of unknown length and therefore there is no solution.
Outcome 2 – there can be only one solution. If the triangle is a right triangle, there is only one
solution. And it is easy to solve using the trig functions.
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Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 6 of 10
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
Outcome 3 – there can be two solutions. For angles between 0° and 180°, the value of the sine
of the angle will always be positive. When the angle is 0° the sine is zero. As the angle
increases, the value of the sine also increases up to a maximum value of 1.000 at 90°. As the
angle continues to increase past 90°, the value of the sine of the angle decreases back down to
zero at 180°. So, when using the value of the sine to get the value of the angle (when using the
Law of Sines), the calculator will not tell you if the angle is less than 90° or more than 90°. The
calculator will return identical values for two angles whose sum is 180° as shown below.
sin 27° = 0.4540 sin 153° = 0.4540 When solving a scalene triangle using the Law of Sines, be careful to use the correct value of the
angle. When you solve for an unknown angle using the sine, for example sin A = 0.4540, the
calculator will probably return a value of 27°. Use your common sense to determine if that is the
correct angle – or not.
Here is an example showing two solutions to a scalene triangle.
Solution No. 1
30
4
0.1250
C = 48.6°
6 0.1250
A = 180° - 48.6° -30°
A = 101.4°
101.4
0.1250
a = 7.84
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Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 7 of 10
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
Solution No. 2
30
4
0.1250
6 0.1250
C = 131.4°
A = 180° - 131.4° -30°
A = 18.6°
18.6
0.1250
a = 2.55
That concludes the trigonometric review. Now let’s do a quick review of significant figures and
rounding.
Significant Figures Review
All answers to arithmetic problems are expected to have a precise answer. The difficulty in
applying this simple principle lies in being able to tell the difference between arithmetic
precision and the validity of an answer.
In most real-world problems such as statics and strength of materials, only the first few digits of
an answer are valid or “significant”. This is because dimensions are commonly rounded, loads
are only approximate (often specified in the building codes as maximum or minimums), and real
world connections, bearings, and geometric configurations are usually simplified. Simplification
allows relatively simple calculations to closely approximate the solutions to very complex
mathematical equations that represent the real boundary conditions.
With the hand held calculators we have today, it is tempting to write many figures in the answer
to an arithmetic problem. However, just because the calculator shows eight figures, doesn’t
mean that they are all valid. Here’s the rule:
An answer cannot be more accurate than the least accurate
number used in the statement of the problem.
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Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 8 of 10
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
This principle can be illustrated by computing the area of a 2” diameter circle. Obviously the
answer is approximately 3.1415926536 in2 as shown below.
2"
3.1415926536 2
Now, let’s say that the diameter of the circle is accurate to three figures (2.00”). That means that
the answer cannot be more accurate than three significant figures – which is 3.14 in2.
Here’s why. Because the diameter is accurate to three significant figures – 2.00” – that means
that the actual value of the diameter is in the range of 1.995” to 2.005”. Therefore, the actual
area is somewhere between 3.12595 in2 and 3.15732 in2 as shown in the following two
calculations.
Minimum area
Maximum area
.
.
3.12595
→ 3.13 in2
3.15732
→ 3.16 in2
Rounding these two values to three significant figures means that the actual area of the circle is
between 3.13 in2 and 3.16 in2. In fact, the area of the circle could actually be 3.13 in2, 3.14 in2,
3.15 in2, or 3.16 in2. Originally, we said that the answer was approximately 3.14 in2 which is
still true – we have not contradicted the principle that the answer cannot be more accurate than
the least accurate number used in the calculation.
Rounding Review
Most values in structural mechanics problems are known with two or three figures of accuracy.
It follows that most answers in statics and strength of materials should have two or three figures
of accuracy. The widely accepted practice is that all answers have three significant figures.
There are a couple of simple exceptions to the three significant figure rule.
 Intermediate values of a calculation can have any number of figures. You are encouraged
to carry extra figures through the calculation process, then round the answer.
 Whole number answers need not have the added zeros. An answer of 5 ft is preferred to
5.00 ft, but both are acceptable.
 When two numbers must add to a fixed total, one number may have an added significant
figure or the other may have one less significant figure. The fixed total has three
significant figures.
o Example:
6.67 + 13.33 = 20.0
OR
6.7 + 13.3 = 20.0
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Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 9 of 10
Basic Trigonometry, Significant Figures, and Rounding – A Quick Review
A SunCam online continuing education course
When rounding an answer to three significant figures, one occasionally meets the dilemma of a
fourth digit that is exactly equal to 5. Should you round up or round down? Both answers are
acceptable, but the widely accepted rule is to round to the even number.
Example: Round the answer 1225 to three significant figures.
Using the widely accepted rule, the answer should be rounded to 1220
instead of 1230.
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Copyright 2014 Prof. Patrick L. Glon, P.E.
Page 10 of 10